Few British bands have had as much impact as the Clash. Blazing out of the London punk scene in the 1970s, they sought nothing less than to change the world and, for many of their fabs, they managed it. In fashion, their early spray-painted outfits inspired bands from punk peers to the Manic Street Preachers, while their later dark-suits-and-hats look was adopted by scores of louche young musicians, notably the Libertines. Politically, they made being committed seem not just acceptable but essential to scores of young musicians. However, their biggest contribution was musical. They were looking to leave punk behind almost from their inception, incorporating reggae, dub, rockabilly, ska, funk and later hip-hop and electro, making quantum leap after quantum leap. Unlike many of their peers, their impact was international, as they sold millions of records and played Shea Stadium in the US withthe Who, who were among their many admirers. Although virtually every track on their debut album has a justifiable claim to be included here, this non-album 1977 single is the high watermark of the Clash’s punk period. And yet, even as the punk fires burned, the song hinted at the dizzying musical adventure to come. In a very shrewd move, it was produced by Lee “Scratch” Perry, the Jamaican reggae legend who had also co-written Junior Murvin’s Police and Thieves, which the Clash covered on that first album. Complete Control’s lyrical tirade against their label, CBS is not exactly revolutionary – the Sex Pistols did a similar thing with EMI – but the passion still raises goose pimples, as does the song’s hurtling sense of romanticism and drama (not least, Mick Jones’s blistering guitar solo) as the Clash are found in a familiar position: cornered but coming out guns blazing, like Paul Newman and Robert Redford at the end of Butch Cassidy and the Sundance Kid.

Clash city rockers – Mick Jones and Paul Simonon recall the glory days

This brilliant 1978 hit encapsulates the Clash’s ability to cover many bases simultaneously, yet at the time it was met with some confusion even among the band’s own fans. “We were a big fat riff group,” Strummer later explained. “We weren’t supposed to do things like that.” What he means is how the Clash incorporated reggae into punk and rock so seamlessly that it became a natural fit, while the song contains what could be the singer’s most complete lyric. It was inspired by a trip to a reggae all-nighter at the Hammersmith Palais, during which Strummer felt that the performances by Dillinger, Leroy Smart and Delroy Wilson were more lightweight than the roots reggae he’d expected. From this kernel of disgruntlement comes a state of the nation address that takes on everything from the rise of the National Front (“If Adolf Hitler were in England today, they’d send a limousine anyway”) to racism (“white youth, black youth, better find a solution”), the distribution of wealth, to pop culture. Along the way, Strummer namechecks Robin Hood, has a pop at some rivals (long believed to be the Jam – “they’ve got Burton suits, huh, you think it’s funny, turning rebellion into money”), then punk’s second wave (“the new groups are not concerned”) and even aims a quip at his own pale, speedy self (“I’m the all-night, drug-prowling wolf, who looks sick in the sun”) One of Strummer’s own favourite Clash songs, it was played at his funeral in 2002.

The Clash’s second album, Give ’Em Enough Rope, was given a cool critical reception – produced by Blue Öyster Cult producer Sandy Pearlman, it was seen as too rock, too American (a big sin for many, given the song I’m So Bored With The USA). The debut had been laid down in London in an amphetamine-fuelled hurry, but this US-recorded follow-up was more painstaking, as Pearlman favoured a more detailed approach, recording take after take and spending an age on each drum sound; Strummer moaned that Pearlman was “trying to turn us into Fleetwood Mac”. The album certainly has its flaws, but contains some real gems, including one of the Clash’s great rock anthems. Ironically, given the band’s embrace of Jamaican music, the song – written in the Pegasus Hotel, Kingston – describes their ill-fated trip to the country, where they didn’t find the promised land of their favourite 12in singles, but a nation riven by crime and gunfire. Thus, “Sitting in my safe European home/ Don’t wanna go back there again.”

Although all the Clash members loved reggae, bassist Paul Simonon was pivotal to the wayJamaican music fired the band. He had grown up amid Brixton’s West Indian community and had been exposed to reggae from early childhood. This 1979 version of Willi Williams’s single from the previous year is the definitive example of the Clash playing reggae. Williams’s version is great – woozier, more ska – but on the B-side of London Calling, the Clash made it their own, from the echoey keyboard intro to the way the groove kicks in over Simonon’s marauding bassline. The lyrics, meanwhile, could have come straight from the Clash’s own songbook – “A lot of people won’t get no justice tonight … so a lot of people are going to have to stand up and fight” – while the sound of fireworks and bombs and Strummer’s seeming ad libs further embellish the song’s smouldering, simmering atmosphere: “OK, OK, don’t push us when we’re hot!” This shouted instruction to the engineers was prompted by the studio clock approaching the three-minute mark – the time agreed that the song would end. It was sounding so good, though, that the band just kept playing.

Already we’ve heard the Clash as a punk band, reggae band and rock band, but this track from 1979’s double album, London Calling, shows just what they could do with a handful of hooks and a pop melody. As ever, the interplay between Strummer and Jones’s combined vocals produces some wonderful two-part harmonies, while the lyrics compare the experiences of freedom fighters fighting fascism in the Spanish civil war with modern-day tourist visitors. As with so many great Clash songs, the lyrics are an education and Strummer excels himself by singing in what he called “Clash Spanish”, but one doesn’t have to be versed or even interested in political history to recognise that it’s an absolute gem of a tune.

Topper Headon is often remembered for being sacked from the band owing to his heroin abuse. which led to him, some years later, selling his gold and platinum discs to fund a raging habit. However, as Strummer would remind anybody, the drummer’s musical contribution to the Clash was Herculean enough to far outweigh such negative aspects towards the end, and went far above the normal for a drummer in a rock band. Although this third single from London Calling is one of Mick Jones’s pop songs, it’s Headon’s superlative work behind the drum stool which takes it into another dimension. His killer drum intro lays down the terrific groove which Jones would later compare to a train – hence the title – and is typical of the kind of deceptively complex, swinging playing that the American novelist Scott Kenemore, also a drummer, once described as “an undiscovered treasure for many”. Happily, Headon eventually recovered and, in 2008, made an emotional return to the stage with Jones, who was then fronting Carbon/Silicon, to play this song together for the first time in 25 years. The song’s ability to travel between musical boundaries is perfectly illustrated by an unlikely cover by country star Dwight Yoakam.

Such was the Clash’s musical output that they could afford to release some of their best songs as singles only, and still have enough in the locker for their albums. This August 1980 seven-inch illustrates the point: originally available only on import, it ended up reaching No 12 in the charts. Bankrobber finds the Clash at their most musically undefinable. Produced by Mikey Dread, it’s perhaps closer to dub than anything, and features a fine Strummer yarn about how his father was a bank robber (“he never hurt nobody”) amid a wider point about how capitalism forces people to grovel. The accompanying video alternates between footage of the band in the studio and comical clips of Clash crew members Barry “The Baker” Auguste and Johnny Green mocking up a bank heist, complete with bandanas over their faces that proved so realistic that they were stopped by the police.

Released just three months after Bankrobber, this November 1980 single from the Sandinista! triple album isn’t the most widely heard of the Clash’s hits, but illustrates the band’s ability to craft a brilliantly genre-busting anthem with a powerful message. The music begins with a siren and the US Marines’ military cadence, and the lyrics reference the University of Chicago Doomsday clock, which supposedly indicates the proximity of the world to nuclear disaster. Mostly, though, as the title indicates, it’s a song about the draft, written soon after President Jimmy Carter had reintroduced selective service registration – the precursor to calling up adult males to the armed forces – in response to the Soviet invasion of Afghanistan. The lyrics “It’s up to you not to heed the call up / I don’t wanna die / I don’t wanna kill” are as straightforward and direct as Career Opportunities, London’s Burning or any of their great punk era protest songs.

By this point, the Clash’s musical adventure was going full steam ahead, and their music shifted up yet another gear when they started assimilating the sounds they were hearing coming out of the Bronx, as they recorded Sandinista! in New York. At this point, few if anyone in the UK had heard of rap or hip-hop, never mind the names of early pioneers such as Grandmaster Flash, the Sugarhill Gang or Afrika Bambaataa. However, Mick Jones became so fascinated with the rapidly emerging music that he took to carrying round a beat box to blast out hip-hop sounds. Just as they had seamlessly incorporated reggae, The Magnificent Seven is Strummer’s take on rap, in the form of a tirade against the cost of capitalism with plenty of quipped, funny detours (“Italian mobster shoots a lobster / Sea food restaurant gets out of hand”). Arguments still rage over whether this or Blondie’s Rapture – recorded a month or two later, but released three months earlier) is the first white rap hit, but the song, written and recorded inside two hours, is undoubtedly an enduring moment. Incidentally, this is one of the few Clash recordings that doesn’t feature Paul Simonon – he was elsewhere, so the nimble bassline was played by Norman Watt-Roy, more usually found in Ian Dury’s Blockheads.

The Clash’s catalogue is so mammoth and so wide-ranging that even 10 tunes barely scratches the surface, and I’ve left out the likes of London Calling (so familiar everyone has surely heard it by now), White Riot, Rock the Casbah and some lesser known gems such as Police on My Back, Somebody Got Murdered, Gates of the West or their fantastic remodel of Toots and the Maytals’ Pressure Drop. This track from 1982’s Combat Rock proved impossible to ignore. Straight To Hell again bears the wonderful signature of Topper Headon, who – in one of his last recordings with the group – lays down a memorable bossa nova beat, while Strummer played the bass drum with a lemonade bottle. By this point, the band were imploding, as musical differences, drug problems and personal disagreements – coupled with the pressure of five years in the public eye – started to take a toll. It wasn’t quite the end, a Jones/Headon-free line-up made 1985’s Cut the Crap, but Combat Rock saw the classic line-up bow out with a bang on their bestselling and arguably most experimental album. Nothing in the canon sounds quite like this haunting revenge anthem, as recently sampled by MIA, a blast at at American soldiers in Vietnam who left local women pregnant. Strummer pulls out one of his more mournful vocals for the otherwise scathing lyrics: “Let me tell you ’bout your blood, bamboo kid / It ain’t Coca-Cola, it’s rice / Straight to hell / Go straight to hell boys.” Strummer once declared the song to be their “absolute masterpiece”: they’d certainly travelled a long, long way from the speed-snorting quartet who started life supporting the Sex Pistols to 50 people in the Black Swan pub in Sheffield and who had declared, “We’re a garage band … we come from garageland.” What a band they were.

• This article was amended on 24 September 2015. An earlier version misnamed a Clash crew member in the video for Bankrobber as Garry Baker. This has been corrected to say Barry “The Baker” Auguste.